On functions convex in the direction of the imaginary axis with real coefficients

Let Υ be a subclass of the class of all analytic functions in the unit disk Δ having the normalization f(0) = f′(0) − 1 = 0. If there exists an analytic, univalent function m satisfying the following conditions: m′ (0) > 0, ∧f ∈Υ m ‹ f and for every analytic function k, k(0) = 0, there is (∧f∈Υ k ‹ f) ⇒ k ‹ m, then this function is called the minorant of Υ. Similarly, if there exists an analytic, univalent function M such that M′ (0) > 0, ∧f ∈Υ f ‹ M and for every analytic function k, k(0) = 0, there is (∧f ∈Υ f ‹ k) ⇒ M ‹ k, then this function is called the majorant of Υ. It is possible to give a number of examples of classes of analytic functions for which the majorant or minorant does not exist. However, if these functions exist then m(Δ) and M(Δ) coincide with the Koebe domain and the covering domain for Υ, respectively. In this paper we determine the Koebe domain and the covering domain as well as the minorant and the majorant for the class consisting of functions convex in the direction of the imaginary axis with real coefficients.